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Theorem 4oa 1039
 Description: Variant of proper 4-OA. (Contributed by NM, 29-Dec-1998.)
Hypotheses
Ref Expression
4oa.1 e = (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
4oa.2 f = (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e)
Assertion
Ref Expression
4oa ((a1 d) ∩ f) ≤ (b1 d)

Proof of Theorem 4oa
StepHypRef Expression
1 4oa.2 . . 3 f = (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e)
21lan 77 . 2 ((a1 d) ∩ f) = ((a1 d) ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e))
3 axoa4a 1037 . . . 4 (((b 1 d) →1 d) ∩ ((b 1 d) ∪ ((a 1 d) ∩ ((((b 1 d) ∩ (a 1 d)) ∪ (((b 1 d) →1 d) ∩ ((a 1 d) →1 d))) ∪ ((((b 1 d) ∩ (c 1 d)) ∪ (((b 1 d) →1 d) ∩ ((c 1 d) →1 d))) ∩ (((a 1 d) ∩ (c 1 d)) ∪ (((a 1 d) →1 d) ∩ ((c 1 d) →1 d)))))))) ≤ ((((b 1 d) ∩ d) ∪ ((a 1 d) ∩ d)) ∪ ((c 1 d) ∩ d))
4 id 59 . . . 4 (b 1 d) = (b 1 d)
5 id 59 . . . 4 (a 1 d) = (a 1 d)
6 id 59 . . . 4 (c 1 d) = (c 1 d)
73, 4, 5, 6oa4to4u2 974 . . 3 ((b1 d) ∩ ((b 1 d) ∪ ((a 1 d) ∩ ((((b1 d) ∩ (a1 d)) ∪ ((b 1 d) ∩ (a 1 d))) ∪ ((((b1 d) ∩ (c1 d)) ∪ ((b 1 d) ∩ (c 1 d))) ∩ (((a1 d) ∩ (c1 d)) ∪ ((a 1 d) ∩ (c 1 d)))))))) ≤ d
8 4oa.1 . . 3 e = (((ac) ∪ ((a1 d) ∩ (c1 d))) ∩ ((bc) ∪ ((b1 d) ∩ (c1 d))))
97, 8oa4uto4g 975 . 2 ((a1 d) ∩ (((ab) ∪ ((a1 d) ∩ (b1 d))) ∪ e)) ≤ (b1 d)
102, 9bltr 138 1 ((a1 d) ∩ f) ≤ (b1 d)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-4oa 1033 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  4oaiii  1040
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